Gauge symmetries and covariant derivatives
Definition: Gauge symmetries
Symmetries whose parametrization $\alpha(x)$ depends on $x$ are called gauge or local symmetries.
Definition: Global symmetries
Symmetries whose parametrization is a constant $\alpha$ not depending on $x$ are called global symmetries.
A global symmetry implies a conserved current according to Noethers theorem.
We can impose gauge symmetry on a Lagrangian containing a vector field $A_\mu(x)$ as it will guarantee the right amount of degrees of freedom for our Representations of the Poincare group.
If we want to construct an interacting theory with e.g. a scalar field $\phi$ all other terms need to respect the gauge symmetry as well.
For that we need a complex scalar field $$\phi(x) = \phi_1(x) + i\phi_2(x)$$ which transforms as $$\phi(x) \to e^{-i\alpha(x)}\phi(x)$$ under gauge transformation. To make the kinetic term $\partial_\mu \phi \partial^\mu \phi$ gauge invariant we introduce the Covariant derivative
Definition: Covariant derivative
The covariant derivative $D_\mu$ is defined as a derivative which leaves the kinetic terms of a theory gauge invariant. For a theory with a vector field $A_\mu$ and a complex scalar $\phi$ it is defined as $$D_\mu \phi = (\partial_\mu + ieA_\mu)\phi,$$ with the charge of $\phi$ as $e$.
This leads to scalar QED
Definition: Scalar QED
Scalar QED is a theory with a massless vector $A_\mu$ and a massive complex scalar $\phi$: $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^(D^\mu\phi) - m^2\phi^\phi$$ It is gauge invariant under $$\phi(x) \to e^{-i\alpha(x)}\phi(x), \quad A_\mu(x) \to A_\mu(x) + \frac{1}{e}\partial_\mu \alpha(x).$$